Methane $\left(\mathrm{CH}_4\right)$ at $25^{\circ} \mathrm{C}, 1 \mathrm{~atm}$ enters a furnace operating at steady state and burns completely with $140 \%$ of theoretical air entering at $400 \mathrm{~K}, 1\ \text{atm}$. The products of combustion exit at $700 \mathrm{~K}, 1 \mathrm{~atm}$. Kinetic and potential energy effects are negligible. If the rate of heat transfet from the furnace to the surroundings is $400 \mathrm{~kW}$, determine the mass flow rate of methane, in $\mathrm{kg} / \mathrm{s}$.

A frame ABC is supported in part by cable DBE that passes through a frictionless ring at B. Knowing that the tension in the cable is 385 N, determine the components of the force exerted by the cable on the support at D.

At a certain point in a pipe, air flows steadily with a velocity of 150 m/s and has a static pressure of $70\ \mathrm{kPa}$ and a static temperature of $4^{\circ} \mathrm{C}$. The flow is adiabatic and frictionless. (a) Determine the maximum possible reduction in area and the following quantities for that minimum area: stagnation pressure, stagnation temperature, static pressure, static temperature, velocity, and Mach number. (b) Determine the quantities listed in (a) at a point where the area is $15 \%$ smaller than the initial area.

A $9~\mathrm{kg}$ uniform disk is attached to the $5~\mathrm{kg}$ slender rod $A B$ by means of frictionless pins at $B$ and $C$. The assembly rotates in a vertical plane under the combined effect of gravity and of a couple $\mathbf{M}$ that is applied to rod $A B$. Knowing that at the instant shown the assembly has an angular velocity of $6 \mathrm{~rad} / \mathrm{s}$ and an angular acceleration of $25 \mathrm{~rad} / \mathrm{s}^2$, both counterclockwise, determine $(a)$ the couple $\mathbf{M},(b)$ the force exerted by pin $C$ on member $A B$

An 18-tooth pinion having a 20 degrees pressure angle meshes with a 36 -tooth gear. The center distance is 10 in. The pinion has stub teeth. The gear has full-depth involute teeth. Determine the contact ratio (tooth intervals in contact) and the diametral pitch.

A 21-tooth spur pinion mates with a 28-tooth gear. The diametral pitch is 3 teeth/in and the pressure angle is $20^{\circ}$. Make a drawing of the gears showing one tooth on each gear. Find and abulate the following results: the addendum, dedendum, clearance, circular pitch, tooth thickness and base-circle diameters; the lengths of the arc of approach, recess, and action; and the base pitch and contact ratio.

Determine the shrink rule to be used by mold makers for die casting of zinc. Using the shrinkage value in the above Table, express your answer in terms of decimal $\mathrm{mm}$ of elongation per $300 \mathrm{~mm}$ of length compared to a standard $300-\mathrm{mm}$ scale.

A long bar of rectangular cross section is $60 \mathrm{mm} \times 90 mm$ on a side and has a thermal conductivity of $1 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$. One surface is exposed to a convection process with air at 100°C and a convection coefficient of $100 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, while the remaining surfaces are maintained at 50 C. (a) Using a grid spacing of 30 mm and the Gauss-Seidel iteration method, determine the nodal temperatures and the heat rate per unit length normal to the page into the bar from the air. (b) Determine the effect of grid spacing on the temperature field and heat rate. Specifically, consider grid spacing of 15 mm. For this grid, explore the effect of changes in h on the temperature field and the isotherms.

An inverted hollow cylinder is pushed into the water as is shown . Determine the distance, $\ell$, that the water rises in the cylinder as a function of the depth, d, of the lower edge of the cylinder. Plot the results for $0 \leq d \leq H$, when H is equal to $1 \mathrm{~m}$. Assume the temperature of the air within the cylinder remains constant.

A $3200-\text{lb}$ forklift truck is used to lift a $1700-\text{lb}$ crate. Determine the reaction at each of the two $(a)$ front wheels $A,(b)$ rear wheels $B$.

The bolt has a diameter of $20 \mathrm{~mm}$ and passes through a tube that has an inner diameter of $50 \mathrm{~mm}$ and an outer diameter of $60 \mathrm{~mm}$. If the bolt and tube are made of A-36 steel, determine the normal stress in the tube and bolt when a force of $40 \mathrm{kN}$ is applied to the bolt. Assume the end caps are rigid.

A uniform sphere of mass M and radius R and a uniform cylinder of mass M and radius R are released simultaneously from rest at. the top of an inclined plane. Which body reaches the bottom first if they both roll without slipping?

Air (1 atm) entered into a 5-mm diameter circular tube at an average velocity of 5 m/s. The tube wall is maintained at a constant surface temperature. Determine the convection heat transfer coefficient for (a) a 10-cm long tube and (b) a 50-cm long tube. Evaluate the air properties at $50^\circ C.$

Determine the location e of the shear center, point O, for the thin-walled member having the cross section shown. The member segments have the same thickness t.